R Under development (unstable) (2025-01-06 r87534 ucrt) -- "Unsuffered Consequences"
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> require("fitdistrplus")
Loading required package: fitdistrplus
Loading required package: MASS
Loading required package: survival
> #We choose a low number of bootstrap replicates in order to satisfy CRAN running times constraint.
> #For practical application, we recommend to use nbboot=501 or nbboot=1001.
> 
> nbboot <- 1001
> nbboot <- 11
> 
> nsample <- 100
> nsample <- 10
> 
> visualize <- FALSE # TRUE for manual tests with visualization of results
> 
> # (1) Fit of a gamma distribution to serving size data
> # using default method (maximum likelihood estimation)
> # followed by parametric bootstrap
> #
> data(groundbeef)
> serving <- groundbeef$serving
> f1 <- fitdist(serving, "gamma")
> b1 <- bootdist(f1, niter=nbboot, silent=TRUE)
> b1 <- bootdist(f1, niter=nbboot, silent=FALSE)
> 
> print(lapply(b1, head))
$estim
     shape       rate
1 3.518018 0.04513514
2 3.870860 0.05274972
3 4.085907 0.05277592
4 4.044606 0.05533558
5 3.936531 0.05093270
6 3.667763 0.04974809

$converg
[1] 0 0 0 0 0 0

$method
[1] "param"

$nbboot
[1] 11

$CI
          Median       2.5%      97.5%
shape 3.93653069 3.53485812 4.68797900
rate  0.05274972 0.04551202 0.06430244

$fitpart
$fitpart$estimate
     shape       rate 
4.00955898 0.05443907 

$fitpart$method
[1] "mle"

$fitpart$sd
      shape        rate 
0.341451641 0.004937239 

$fitpart$cor
          shape      rate
shape 1.0000000 0.9384578
rate  0.9384578 1.0000000

$fitpart$vcov
            shape         rate
shape 0.116589223 1.582079e-03
rate  0.001582079 2.437633e-05

$fitpart$loglik
[1] -1253.625


> plot(b1)
> summary(b1)
Parametric bootstrap medians and 95% percentile CI 
          Median       2.5%      97.5%
shape 3.93653069 3.53485812 4.68797900
rate  0.05274972 0.04551202 0.06430244
> 
> # (2) new plot arguments
> #for new graph functions
> f1 <- fitdist(rgamma(nsample, 2, 3), "gamma")
> b1 <- bootdist(f1, niter=nbboot, silent=TRUE)
> plot(b1)
> plot(b1, trueval = c(2, 3))
> plot(b1, enhance=TRUE)
> plot(b1, enhance=TRUE, trueval = c(2, 3))
> plot(b1, enhance=TRUE, rampcol=c("blue", "green"), nbgrid=15, nbcol=15)
> 
> if(any(installed.packages()[, "Package"] == "actuar") && visualize)
+ {
+   require("actuar")
+   set.seed(123)
+   f1 <- fitdist(rburr(nsample, 2, 3, 1), "burr", start=list(shape1=10, shape2=10, rate=1))
+   b1 <- bootdist(f1, niter=nbboot, silent=TRUE)
+   plot(b1)
+   plot(b1, trueval = c(2, 3, 1))
+   plot(b1, enhance=TRUE)
+   plot(b1, enhance=TRUE, trueval = c(2, 3, 1))
+ }
> 
> 
> # (3) estimation of the rate of a gamma distribution 
> # by maximum likelihood with the shape fixed at 4 using the argument fix.arg
> # followed by parametric bootstrap
> #
> f1c  <- fitdist(serving, "gamma", start=list(rate=0.1), fix.arg=list(shape=4))
> b1c  <-  bootdist(f1c, niter=nbboot)
> summary(b1c)
Parametric bootstrap medians and 95% percentile CI 
    Median       2.5%      97.5% 
0.05488348 0.05337624 0.05656264 
> 
> # (4) fit of a gamma distribution to serving size data 
> # by quantile matching estimation (in this example matching 
> # first and third quartiles) followed by parametric bootstrap
> #
> f1d  <-  fitdist(serving, "gamma", method="qme", probs=c(0.25, 0.75))
> b1d  <-  bootdist(f1d, niter=nbboot)
> summary(b1d)
Parametric bootstrap medians and 95% percentile CI 
          Median       2.5%      97.5%
shape 4.31100756 3.26207478 4.86018255
rate  0.05665868 0.04148652 0.06415661
> 
> # (5) fit of a gamma distribution with control of the optimization
> # method,  followed by parametric bootstrap
> #
> if(visualize) { # check ERROR on aarch64-apple-darwin20.4.0 (64-bit) (2021/05/12)
+   set.seed(1234)
+   f1e <- fitdist(serving, "gamma", optim.method="L-BFGS-B", lower=c(0, 0))
+   b1e <- bootdist(f1e, niter=nbboot)
+   summary(b1e)
+ }
> 
> # (6) fit of a discrete distribution by matching moment estimation 
> # (using a closed formula) followed by parametric bootstrap
> #
> set.seed(1234)
> x2 <- rpois(nsample, lambda = 5)
> f2 <- fitdist(x2, "pois", method="mme")
> b2 <- bootdist(f2, niter=nbboot)
> plot(b2,pch=16)
> summary(b2)
Parametric bootstrap medians and 95% percentile CI 
Median   2.5%  97.5% 
 4.400  2.975  4.900 
> 
> # (7) Fit of a uniform distribution using the Cramer-von Mises distance
> # followed by parametric bootstrap 
> # 
> if(visualize)
+ {
+   x3  <-  runif(nsample, min=5, max=10)
+   f3  <-  fitdist(x3, "unif", method="mge", gof="CvM")
+   b3  <-  bootdist(f3, bootmethod="param", niter=nbboot)
+   summary(b3)
+   plot(b3)
+ }
> 
> # (9) fit of a Weibull distribution to serving size data by maximum likelihood 
> # estimation or by quantile matching estimation (in this example matching 
> # first and third quartiles) followed by parametric bootstrap
> #
> fWmle <- fitdist(serving, "weibull")
> bWmle <- bootdist(fWmle, niter=nbboot)
> summary(bWmle)
Parametric bootstrap medians and 95% percentile CI 
         Median      2.5%     97.5%
shape  2.161366  2.062089  2.380348
scale 84.435931 77.331984 85.643448
> quantile(bWmle, probs=c(0.25, 0.75))
(original) estimated quantiles for each specified probability (non-censored data)
           p=0.25  p=0.75
estimate 47.13642 96.7809
Median of bootstrap estimates
           p=0.25   p=0.75
estimate 46.97044 97.32588

two-sided 95 % CI of each quantile
         p=0.25   p=0.75
2.5 %  42.98426 90.03661
97.5 % 50.73265 99.75572
> 
> fWqme <- fitdist(serving, "weibull", method="qme", probs=c(0.25, 0.75))
> bWqme <- bootdist(fWqme, niter=nbboot)
> summary(bWqme)
Parametric bootstrap medians and 95% percentile CI 
         Median      2.5%     97.5%
shape  2.249725  2.143962  2.418769
scale 86.476953 84.002435 89.635212
> quantile(bWqme, probs=c(0.25, 0.75))
(original) estimated quantiles for each specified probability (non-censored data)
          p=0.25   p=0.75
estimate 50.0001 99.99983
Median of bootstrap estimates
           p=0.25   p=0.75
estimate 49.72662 99.90189

two-sided 95 % CI of each quantile
         p=0.25    p=0.75
2.5 %  47.93772  97.18378
97.5 % 53.34119 102.88616
> 
> 
> # (10) Fit of a Pareto distribution by numerical moment matching estimation
> # followed by parametric bootstrap
> # !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! LONG TO RUN !!!!!!!!!!!!!!!!!!!!!!!!
> #
> if (visualize) 
+ {
+   if(any(installed.packages()[, "Package"] == "actuar"))
+   {
+     require("actuar")
+     #simulate a sample
+     x4 <- rpareto(nsample,  6,  2)
+     memp <- function(x,  order)
+       ifelse(order == 1,  mean(x),  sum(x^order)/length(x))
+     
+     f4 <- fitdist(x4,  "pareto",  "mme",  order=1:2,  
+                   start=list(shape=10,  scale=10),  
+                   lower=1,  memp="memp",  upper=50)
+     
+     b4 <- bootdist(f4,  niter=nbboot)
+     summary(b4)
+     
+     b4npar <- bootdist(f4,  niter=nbboot,  bootmethod="nonparam")
+     summary(b4npar)
+   }
+ }
> 
> # (11) Fit of a Burr distribution (3 parameters) using MLE 
> # followed by parametric boostrap
> # !!!!!!!!!!!!!!!! LONG TO RUN !!!!!!!!!!!!!!!!!!
> #
> if (visualize)
+ {
+   if(any(installed.packages()[, "Package"] == "actuar"))
+   {
+     require("actuar")
+     data(danishuni)
+     
+     fdan <- fitdist(danishuni$Loss, "burr", method="mle", 
+                     start=list(shape1=5, shape2=5, rate=10), lower=0+1e-1, control=list(trace=0))
+     bdan <- bootdist(fdan,  bootmethod="param", niter=nbboot)
+     summary(bdan)
+     plot(bdan)
+     cdfcomp(fdan, xlogscale=TRUE)
+   }
+ }
> 
> # (12) Fit of a Triangular distribution (3 parameters) using MLE 
> # followed by parametric boostrap, with crashes of optim
> # 
> if(any(installed.packages()[, "Package"] == "mc2d"))
+ {
+   require("mc2d")
+   set.seed(1234)
+   x4 <- rtriang(100,min=0,mode=4,max=20) # nsample not used : does not converge if the sample is too small
+   fit4t<-fitdist(x4,dtriang,start=list(min=0,mode=4,max=20))
+   summary(fit4t)
+   b4t<-bootdist(fit4t,niter=nbboot) 
+   b4t
+   plot(b4t)
+   summary(b4t)
+   quantile(b4t)
+   
+ }
Loading required package: mc2d
Loading required package: mvtnorm

Attaching package: 'mc2d'

The following objects are masked from 'package:base':

    pmax, pmin

(original) estimated quantiles for each specified probability (non-censored data)
           p=0.1    p=0.2   p=0.3    p=0.4    p=0.5   p=0.6    p=0.7    p=0.8
estimate 6.99066 7.110847 7.20307 7.280817 7.349314 7.41124 7.468186 7.522676
            p=0.9
estimate 7.590967
Median of bootstrap estimates
            p=0.1    p=0.2    p=0.3    p=0.4    p=0.5    p=0.6   p=0.7    p=0.8
estimate 7.381901 7.411768 7.439453 7.462793 7.482165 7.500423 7.51904 7.534952
            p=0.9
estimate 7.552589

two-sided 95 % CI of each quantile
          p=0.1    p=0.2    p=0.3    p=0.4    p=0.5    p=0.6    p=0.7    p=0.8
2.5 %  7.312251 7.340770 7.360830 7.378643 7.395703 7.414393 7.435609 7.460138
97.5 % 7.516127 7.529222 7.543164 7.558143 7.574432 7.592449 7.612900 7.637158
          p=0.9
2.5 %  7.489453
97.5 % 7.670650

The estimation method converged only for 9 among 11 bootstrap iterations.
> 
> # (13) Fit of a Pareto and a Burr distribution, with bootstrap on the Burr distribution
> #
> #
> if(visualize)
+ {
+   data(endosulfan)
+   ATV <-endosulfan$ATV
+   plotdist(ATV)
+   descdist(ATV,boot=nbboot)
+   
+   fln <- fitdist(ATV, "lnorm")
+   summary(fln)
+   gofstat(fln)
+   
+   # use of plotdist to find good reasonable initial values for parameters
+   plotdist(ATV, "pareto", para=list(shape=1, scale=500))
+   fP <- fitdist(ATV, "pareto", start=list(shape=1, scale=500))
+   summary(fP)
+   gofstat(fP)
+   
+   # definition of the initial values from the fit of the Pareto
+   # as the Burr distribution is the Pareto when shape2 == 1
+   fB <- fitdist(ATV, "burr", start=list(shape1=0.3, shape2=1, rate=1))
+   summary(fB)
+   gofstat(fB)
+   
+   cdfcomp(list(fln,fP,fB),xlogscale=TRUE)
+   qqcomp(list(fln,fP,fB),xlogscale=TRUE,ylogscale=TRUE)
+   ppcomp(list(fln,fP,fB),xlogscale=TRUE,ylogscale=TRUE)
+   denscomp(list(fln,fP,fB)) # without great interest as hist does accept argument log="x"
+   
+   # comparison of HC5 values (5 percent quantiles)
+   quantile(fln,probs=0.05)
+   quantile(fP,probs=0.05)
+   quantile(fB,probs=0.05)
+   
+   # bootstrap for the Burr distribution
+   bfB <- bootdist(fB,niter=nbboot)
+   plot(bfB)
+ }
> 
> # (14) relevant example for zero modified geometric distribution
> #
> dzmgeom <- function(x, p1, p2)
+ {
+   p1 * (x == 0) + (1-p1)*dgeom(x-1, p2)
+ }
> pzmgeom <- function(q, p1, p2)
+ {
+   p1 * (q >= 0) + (1-p1)*pgeom(q-1, p2)
+ }
> rzmgeom <- function(n, p1, p2)
+ {
+   u <- rbinom(n, 1, 1-p1) #prob to get zero is p1
+   u[u != 0] <- rgeom(sum(u != 0), p2)+1
+   u
+ }
> 
> x2 <- rzmgeom(nsample, 1/2, 1/10)
> 
> f2 <- fitdist(x2, "zmgeom", method="mle", fix.arg=function(x) list(p1=mean(x == 0)), start=list(p2=1/2))
> b2 <- bootdist(f2, niter=nbboot)
> plot(b2)
> 
> f3 <- fitdist(x2, "zmgeom", method="mle", start=list(p1=1/2, p2=1/2))
> b3 <- bootdist(f3, niter=nbboot)
> plot(b3, enhance=TRUE)
> 
> # (15) does fixing p1 reduce bias of estimating p2?
> summary(b2$estim[, "p2"] - 1/10)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.03889 0.08847 0.12500 0.13867 0.16236 0.31176 
> summary(b3$estim[, "p2"] - 1/10)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.04813 0.11748 0.12577 0.15418 0.19396 0.28762 
> 
> par(mfrow=c(1, 2))
> hist(b2$estim[, "p2"] - 1/10, breaks=100, xlim=c(-.015, .015))
> hist(b3$estim[, "p2"] - 1/10, breaks=100, xlim=c(-.015, .015))
> par(mfrow=c(1, 1))
> 
> # (16) efficiency of parallel operation
> if (visualize)
+ {
+   niter <- 1001 
+   data(groundbeef)
+   serving <- groundbeef$serving
+   f1 <- fitdist(serving, "gamma")
+   
+   alltime <- matrix(NA, 9, 5)
+   colnames(alltime) <- c("user.self",  "sys.self",   "elapsed",    "user.child", "sys.child" )
+   rownames(alltime) <- c("base R", paste("snow", 1:4), paste("multicore", 1:4))
+   
+   alltime[1,] <- system.time(res <- bootdist(f1, niter = niter))
+   
+   for (cli in 1:4)
+   {
+     cat("\nnb cluster", cli, "\n")
+     #ptm <- proc.time()
+     alltime[cli+1,] <- system.time(res <- bootdist(f1, niter = niter, parallel = "snow", ncpus = cli))
+     print(summary(res))
+     #print(proc.time() - ptm)
+     
+   }
+   # not available on Windows
+   if(.Platform$OS.type == "unix") 
+     for (cli in 1:4)
+     {
+       cat("\nnb cluster", cli, "\n")
+       #ptm <- proc.time()
+       alltime[cli+5,] <- system.time(res <- bootdist(f1, niter = niter, parallel = "multicore", ncpus = cli))
+       print(summary(res))
+       #print(proc.time() - ptm)
+       
+     }
+   
+   alltime
+ }
> 
> # (17) bootdist with weights (not yet available, test of error message)
> #
> x <- rpois(nsample, 10)
> xtab <- table(x)
> xval <- sort(unique(x))
> (f1 <- fitdist(x, "pois"))
Fitting of the distribution ' pois ' by maximum likelihood 
Parameters:
       estimate Std. Error
lambda     10.5   1.024695
> (f2 <- fitdist(xval, "pois", weights = xtab))
Fitting of the distribution ' pois ' by maximum likelihood 
Parameters:
       estimate Std. Error
lambda 10.50006   1.024701
Warning message:
In mledist(data, distname, start = arg_startfix$start.arg, fix.arg = arg_startfix$fix.arg,  :
  weights are not taken into account in the default initial values
> summary(bootdist(f1, niter = nbboot))
Parametric bootstrap medians and 95% percentile CI 
  Median     2.5%    97.5% 
10.99923  9.55000 11.90000 
> try(summary(bootdist(f2, niter = nbboot))) # not yet developed
Error in bootdist(f2, niter = nbboot) : 
  Bootstrap is not yet available when using weights
> 
> # (18) density of bootdist()
> #
> x <- rlnorm(1e3)
> b0 <- bootdist(fitdist(x, "lnorm"), niter = 50)
> b1 <- bootdist(fitdist(x, "lnorm"), niter = 100)
> b2 <- bootdist(fitdist(x, "lnorm"), niter = 200)
> 
> #d1 <- fitdistrplus:::density.bootdist(b0, b1, b2)
> d1 <- density(b0, b1, b2)
> str(d1)
List of 3
 $ :List of 2
  ..$ meanlog:List of 11
  .. ..$ x         : num [1:512] -0.078 -0.0776 -0.0772 -0.0768 -0.0764 ...
  .. ..$ y         : num [1:512] 0.00758 0.00835 0.0092 0.01014 0.01114 ...
  .. ..$ bw        : num 0.0126
  .. ..$ n         : int 50
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.0321
  .. ..$ sd        : num 0.0312
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
  ..$ sdlog  :List of 11
  .. ..$ x         : num [1:512] 0.941 0.941 0.941 0.942 0.942 ...
  .. ..$ y         : num [1:512] 0.0124 0.0138 0.0154 0.0171 0.019 ...
  .. ..$ bw        : num 0.0075
  .. ..$ n         : int 50
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 1
  .. ..$ sd        : num 0.0208
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
 $ :List of 2
  ..$ meanlog:List of 11
  .. ..$ x         : num [1:512] -0.0598 -0.0594 -0.0591 -0.0587 -0.0583 ...
  .. ..$ y         : num [1:512] 0.00996 0.01131 0.01278 0.0144 0.01628 ...
  .. ..$ bw        : num 0.0101
  .. ..$ n         : int 100
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.0314
  .. ..$ sd        : num 0.0308
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
  ..$ sdlog  :List of 11
  .. ..$ x         : num [1:512] 0.93 0.93 0.931 0.931 0.931 ...
  .. ..$ y         : num [1:512] 0.0144 0.0161 0.0179 0.0199 0.0221 ...
  .. ..$ bw        : num 0.00724
  .. ..$ n         : int 100
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.995
  .. ..$ sd        : num 0.0202
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
 $ :List of 2
  ..$ meanlog:List of 11
  .. ..$ x         : num [1:512] -0.104 -0.103 -0.103 -0.102 -0.102 ...
  .. ..$ y         : num [1:512] 0.00242 0.00282 0.00327 0.00379 0.00439 ...
  .. ..$ bw        : num 0.00923
  .. ..$ n         : int 200
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.0297
  .. ..$ sd        : num 0.0311
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
  ..$ sdlog  :List of 11
  .. ..$ x         : num [1:512] 0.914 0.915 0.915 0.915 0.915 ...
  .. ..$ y         : num [1:512] 0.00352 0.00407 0.0047 0.0054 0.00618 ...
  .. ..$ bw        : num 0.00656
  .. ..$ n         : int 200
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 1
  .. ..$ sd        : num 0.021
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
 - attr(*, "distname")= chr "lnorm"
 - attr(*, "nbobject")= int 3
 - attr(*, "nbboot")= num [1:3] 50 100 200
 - attr(*, "n")= int 1000
 - attr(*, "class")= chr "density.bootdist"
> plot(d1)
> print(d1)

Bootstrap values for: lnorm for 3 object(s) with 50, 100, 200 bootstrap values (original sample size 1000).> 
> d1 <- density(b1)
> str(d1)
List of 1
 $ :List of 2
  ..$ meanlog:List of 11
  .. ..$ x         : num [1:512] -0.0598 -0.0594 -0.0591 -0.0587 -0.0583 ...
  .. ..$ y         : num [1:512] 0.00996 0.01131 0.01278 0.0144 0.01628 ...
  .. ..$ bw        : num 0.0101
  .. ..$ n         : int 100
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.0314
  .. ..$ sd        : num 0.0308
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
  ..$ sdlog  :List of 11
  .. ..$ x         : num [1:512] 0.93 0.93 0.931 0.931 0.931 ...
  .. ..$ y         : num [1:512] 0.0144 0.0161 0.0179 0.0199 0.0221 ...
  .. ..$ bw        : num 0.00724
  .. ..$ n         : int 100
  .. ..$ old.coords: logi FALSE
  .. ..$ call      : language density.default(x = x[[j]]$estim[, i], bw = bw, adjust = adjust, kernel = kernel)
  .. ..$ data.name : chr "x[[j]]$estim[, i]"
  .. ..$ has.na    : logi FALSE
  .. ..$ mean      : num 0.995
  .. ..$ sd        : num 0.0202
  .. ..$ distname  : chr "lnorm"
  .. ..- attr(*, "class")= chr "density"
 - attr(*, "distname")= chr "lnorm"
 - attr(*, "nbobject")= int 1
 - attr(*, "nbboot")= num 100
 - attr(*, "n")= int 1000
 - attr(*, "class")= chr "density.bootdist"
> plot(d1)
> print(d1)

Bootstrap values for: lnorm for 1 object(s) with 100 bootstrap values (original sample size 1000).> 
> proc.time()
   user  system elapsed 
   6.42    0.48    6.89